Research Seminars @ Illinois

In Higher Algebra, J. Lurie developed a theory of derived centers for algebras over ∞-operads. Similar to how the classical center of an associative k-algebra is a commutative k- algebra, the derived center of an O-algebra is an E1-algebra object in the category of O-algebras. In the case of E1-algebras, the Dunn Additivity Theorem thus promotes the derived center to an E2-algebra. By defining the Hochschild complex of an E1-algebra object as its derived center, we hence obtain a built-in solution of Deligne’s conjecture on Hochschild cochains. We show that for an associative k-algebra, this definition recovers the classical Hochschild complex, including the correct Gerstenhaber algebra structure in cohomology. Globalizing to schemes, we show that the derived E1-center of the structure sheaf is indeed glued from the local centers, and that for a smooth scheme we recover the sheaf of polydifferential operators. The motivation for this work has its origin in Kontsevich’s description of the action of the Grothendieck-Teichmüller group on the Hochschild cohomology of a smooth algebraic variety.